Complexity of unknotting of trivial 2-knots.

Boris Lishak and Alexander Nabutovsky

October, 8, 2015

Abstract.

We construct families of trivial 2-knots Ki in R4
such that the maximal complexity
of 2-knots in any isotopy connecting Ki
with the standard unknot grows faster
than a tower of exponentials of any fixed height of the complexity of Ki.

Here we can either construct Ki as
smooth embeddings and measure their complexity as the ropelength (a.k.a
the crumpledness) or construct PL-knots Ki, consider isotopies through PL
knots,
and measure the complexity of a PL-knot as the minimal number of flat 2-simplices in its triangulation.

These results contrast with the situation of classical knots in R3,
where every unknot can be untied through knots of complexity that is only
polynomially higher than the complexity of the initial knot.

1. Main result.

Let k be a PL-unknot in R3 with N crossings
on one of its plane projection. The results of [Dyn] imply that k can be isotoped to the standard unknot through PL-unknots with at most 2(N+1)2 crossings.
(See also [L] for further results in this direction.)
On the other hand it was proven in [NW] that for each n≥3
and each computable function f there exists a trivial knot k:Sn⟶Rn+2 triangulated into N (flat) n-simplices such
that any isotopy between k and the trivial unknot that passes through
PL-knots must pass through a knot that cannot be triangulated into
less than f(N) simplices.

Alternatively, one can consider smooth embeddings of Sn into
Rn+2 (or Sn+2)
and measure the complexity of knots as their ropelength (also known as
crumpledness - see [N]) that was defined as vol(k)1nr(k), where
vol(k) is the volume of k, and r(k) denotes the injectivity
radius of the normal exponential map for k. In other words, r(k) is
the supremum of all x such that any two normals to k of length ≤x
do not intersect. Informally speaking, one can think of r(k) as the
maximal radius of a nonself-intersecting tube centered at k.
For this measure of complexity it will still be true that if n=1,
then there exists a polynomial upper bound for the complexity of knots
in an optimal isotopy connecting an unknot with the standard unknot,
and for n>2 the worst case complexity of knots in the optimal isotopies
grows faster than any computable function.

It is natural to conjecture that the results of [NW] for n>2 will also
hold in the case n=2. Here we will prove that the complexity of untying
of a trivial 2-knot can grow faster than a tower of exponentials of
any fixed height of the complexity of the unknot.

Theorem 1.1.

For each positive k and arbitrarily large N
there exists a trivial 2-knot with
complexity x≥N in R4 or S4 such that
any isotopy between this knot and the standard 2-sphere passes
through 2-knots of complexity
≥22…2x (k times).
(In other words, one needs to increase the complexity more than any tower of exponentials of a fixed height of the initial complexity
before the 2-knots can be untied.)
Here “complexity” means either
the number of flat 2-simplices in a triangulation
of the original PL-knot and each of the intermediate
2-knots (and in this case intermediate unknots also
must be PL), or, if the original knot and intermediate knots
are smooth, the complexity of a knot can be defined as its ropelength,
√Arear, where r is the injectivity radius of the
normal exponential map in the ambient R4.

In order to prove this theorem we first construct a sequence of finite presentations
of Z. These finite presentations have certain algebraic
properties that help to realize them as “visible” finite presentations
of 2-knots of complexity comparable with the total length of the
corresponding finite presentations. Moreover, these finite presentations
have the following additional property: In each of them there exists
a trivial element of length n comparable with the total length
of the presentation, Cn, such that one needs to apply
the relations at least 222…2n (constn times)
in order to demontrate that this element is, indeed, trivial.

Then we prove that
the finiteness/effective compactness of the set of trivial 2-knots
of bounded complexity (modulo the group of transformations of R4 generated by dilations and
translations) implies that if all trivial 2-knots could be “untied” without
a very large increase of complexity, then we would be able to contract any null-homotopic
closed curve in the complement to the original 2-knot to a point through closed curves that are not
much longer than the original one.
But the algebraic property of the finite presentations of
2-knot groups explained in the previous paragraph implies that this is not the case.

2. Finite presentations of the trivial group.

Recall that the group that has the
finite presentation with two generators and one relator
<x1,x2|xx21=x21> is called
the Baumslag-Solitar group. (Here and below we use the standard notation
xy for yxy−1.) Note that for each mx1x2m=x12m,
and
therefore the commutator wm=[x1x2m,x1]=e. However, one needs to apply the relation ∼2m times in order to demonstrate this fact in the most
obvious way.
In fact, it is well-known that there is no essentially shorter
way to write wm as a product of the conjugates of the relator and its inverse. In other words, the Dehn function
of the Baumslag-Solitar group is (at least) exponential. A proof
of this fact can be found in [G] (and a sketch of another simpler proof
using van Kampen diagrams can be found in [S]).
The idea of the proof in the paper of Gersten is that the
Baumslag-Solitar group is an HNN-extension and, therefore, the realization
complex of the finite presentation will be aspherical. (Recall that
the realization
complex of a finite presentation has one 1-dimensional cell
corresponding to each generator of the group and one 2-cell for each relator of the group.) So, its universal covering will be contractible, and, in particular,
will have the trivial second homology group. Therefore, there will be
a unique way to fill each null homologous 1-chain in the universal covering
by a 2-chain. Each way to represent wm as the product of conjugates of
the relator and its inverse corresponds to a filling of the lift
of the loop corresponding to wm in the realization complex to
its universal covering. Therefore, it must have the same number of 2-cells
counted with multiplicities as the filling corresponding to the
obvious presentation of wm
as a product of conjugates of the relator and its inverse.
It remains only to check that 2-cells in the universal
covering that correspond to the obvious representaion of wm as the product
of conjugates of the relator and its inverse do not cancel.
Of course, the same proof implies that for each N any representation
of wNm as a product of conjugates of the relator and
its inverse has at least ∼N2m terms.

One can iterate the idea used in the construction of the Baumslag-Solitar
groups and consider the following sequence
of finite presentation of groups (see [B]). For each n=1,2,…

Gn=<x1,…,xn|xx21=x21,xx32=x22,…,xxnn−1=x2n−1>.

This finite presentation has n generators and n−1 relators of total
length 5(n−1).
One can prove that the Dehn function of Gn grows as
22…2k, where the height of the tower of exponentials is n,
using the approach of [Ger] (alternatively, one can use xi-bands) -see [B].
In particular,
one can consider gk∈Gn defined as xx…xkn21.
It is easy to see that gk=x2…2k1 ((n−1) times).
Therefore, vk=gkx1g−1kx−11 will be trivial. One needs to apply relations
more than 22…2O(k)(n−1)) times to demonstrate that vk is trivial, when one proceeds in the obvious way. As above, one can use
the asphericity of the representation 2-complex to conclude that
the filling in its universal covering is unique on the 2-chain level. Below we will choose k=n ”diagonalizing” this construction.
Again, the same lower bound will hold for
any non-zero power of vn with the same proof.

Now conider the following finite presentations Pn=<Gn,t|tvnt−1=vnxn>=<x1,…xn,t|xx21=x21,…,xxnn−1=x2n−1,tvnt−1=vnxn>, where vn=[xx…xnn21,x1] are
words considered in the previous paragraph. It is easy to see that Pn is the finite presentation
of Z=<t>, as vn=e in the corresponding Gromov group of total length const2n. The following theorem is the key technical fact in our paper.
It asserts that any way to demonstrate that, say, xn=e in Pn would involve at least
22…2 (O(n) times) applications of the relations.
Equivalently, each representation of xn as the product of conjugates of the
relators and their inverses must involve at least this number of terms.
We are stating and proving this fact
using the language of van Kampen diagrams (cf. [LS]).

Theorem 2.1.

Proof.

Consider a minimal van Kampen
diagram with xn on the outer boundary. It must contain cells corresponding
to the last relation. As there are no copies of t on the boundary, these cells must form annuli (t-annuli) (see Figure 1).
Consider one of the innermost t-annuli (that does not have any t-cells inside). Its inner boundary must be a non-zero power
of either vn or vnxn. The second option is impossible, as this would imply that the power of vnxn is trivial
in Gn, which is false. So, we have a non-zero power of vn on the innermost boundary. The part of the van Kampen diagram
inside this innermost boundary is a van Kampen diagram in Gn. But we already established that any such diagram must have
size of at least 22…2n (O(n) times).
∎

Figure 1. A Van Kampen diagram for xn. t-annuli are marked by grey.

3. Construction of a 2-knot.

Note that if one adds one more relator to the finite presentation Pn,
namely, t, then
one obtains a finite presentation of the trivial group. Denote the resulting
finite presentation of the trivial group by Qn. The finite
presentation Qn can be trasformed
to the trivial finite presentation of the trivial group by performing O(n)
elementary operations of the following types: 1) Replacing a relator by its inverse; 2) Replacing a relator by its product with another relator; and
3) Replacing a relator r by grg−1 or g−1rg, where g is a
generator. Indeed, one can use O(n) operations involving only the relators
tvnt−1x−1nv−1n and t to replace these two relators by xn
and t, and then O(1) operations to transform
each of the first n−1 relators to xi for an appropriate i.
(The trivial finite presentation here is the finite presentation, where the set of relators
coincides with the set of generators). Note the Qn, the considered
trivial finite presentation and all intermediate finite presentations
are balanced, that is the number of generators is equal to the number
of relators.

For each balanced finite presentation P of the trivial group we can construct
a smooth 4-manifold in R5 by starting from the connected
sum of several copies of S1×S3, where S1 in each copy of
S1×S3 corresponds to one of the generators, and then performing
surgeries killing the relators. More precisely, we realize each relator by a
simple closed curve γ, remove the tubular neighnorhood of γ,
glue in a copy of D2×S2 so that its boundary
is glued to the boundary
of the removed tubular neighnorhood of γ, so that the
boundary of D2 is
glued to a curve isotopic to γ, and smooth out
the boundary. Alternatively, we could start from the 2-complex with one
0-cell, 1-cells corresponding to the generators and 2-cells corresponding
to relators of P (i.e. the realization complex of P), embed it into
R5, take the boundary of an open neighborhood of P and smooth-out
the corners. The resulting smooth 4-manifold will have the fundamental group with the obvious
finite presentation
P (in particular, it will be isomorphic to the trivial group), and the trivial second homology
group. So, it will be a homotopy 4-sphere. Denote it by M4(P).
But it is easy to see that M4(P) will be diffeomorphic to S4.
The reason is that if two balanced finite presentations P and Q are
related by one elementary operation of any of the three types introduced
in the previous paragraph, the manifolds M4(P) and M4(Q) are
diffeomorphic via a diffeomorphism that can be described as a “handle
slide”. Each elementary operation with finite presentation corresponds
to an isotopy of a curve bounding a 2-disc forming an axis of a 2-handle.
The isotopy of the boundary of a 2-disc can be extended to an isotopy of
the 2-disc, the whole 2-handle and the whole 4-manifold. After
finitely many operations we will end up with the 4-manifold M(Tn)
constructed from a finite presentation Tn
with the same generator as Qn and
relators killing all the generators, which is diffeomorphic to the standard
S4
by means of an obvious diffeomorphism.

Thus, each handle slide as well as the whole sequence of handle slides
used to constract diffeomorphisms between M4(Pn) and M4(Tn) can
be regarded as an isotopy M4t(Qn),t∈[0,1],M40(Qn)=M4(Qn),M41(Qn)=M4(Tn). This isotopy can then be extended to an obious isotopy
between M4(Pn) and the round sphere S4 of radius one.

Before moving further we are going to give the following definition:

Definition 3.1.

Let f and g be two positive valued functions
defined on a closed unbounded subset D of [0,∞). We say that they have
similar growth if there exist N0 such that f(x)<exp(…(exp(g({exp(exp(…x))}D)))) (N0 exponentiations both in the
argument of g and of the value of g) and g(x)<exp(…(exp({f(exp(exp(…x))}D)))) (N0 exponentiations both of x and f). {y}D here means min{x∈D,x≥y}. Increasing functions that do not have similar growth with
f(x)=x (restricted to their domain) are called rapidly growing
functions. An increasing function that is not rapidly growing is called reasonably
growing.

Now note that our explanation of why M(Qn) is diffeomorphic to S4
can be used to construct an explicit diffeomorphism such that its Lipschitz
constant regarded as a function of n is bounded by a reasonably growing function of n.

Next consider the last relator, t, and represent it by a simple curve in M(Qn)
that we will also denote t. It corresponds to a 2-handle Ht in
M(Qn). The 2-disc D filling t forms a generator of this handle,
which is diffeomorphic to S2×D. Consider the 2-sphere Sn=S2×c, where c is a point inside D. We claim that the fundamental group
of the complement M(Qn)∖Sn is isomorphic to Z, and,
in fact, it is that this group has “apparent” finite presentation
Pn.
Indeed, M(Qn)∖Sn can be deformed to M(Pn) minus
a tubular neighborhood of t. Yet the deleted
tubular neighborhood of t decomposes
into a 3-cell and a 4-cell. Therefore, if one attaches the deleted
tubular neighborhood back then the “apparent”
finite presentation of the fundamental group remains unchanged.
Thus, Pn is an “apparent” finite presentation of M(Qn)∖Sn.

The meaning of ”apparent” finite presentation
here is that each loop in M(Qn)∖Sn can be homotoped to a bouquet
of loops representing the generators of Pn with an
insignificant length increase. (Here and below a length increase is regarded as
insignificant if it is measured by a reasonably growing function of n.)

Consider a 2-knot Sn in M(Qn). Now consider a diffeomorphism between M(Qn) and M(Tn) that can be obtained as a sequence of handle slides corresponding to
elementary operations transforming Pn into Tn. Take the composition
of this diffeomorphism with a diffeomorphism between M(Tn) and the standard
round sphere S4. Denote the resulting diffeomorphism
between M(Pn) and the round S4 by ϕn. Note that it is easy ensure that the
Lipschitz constants of ϕn and its inverse were bounded by reasonably growing
functions of n. Indeed, it is sufficient to verify that this will hold for diffeomorphisms
corresponding to the individual handle slides. Now each handle slide can be regarded
as an isotopy extension that extends an isotopy of a simple curve bounding a 2-disc
forming an axis of a 2-handle. One can discretize this isotopy of closed curve into small isotopes where the closed curves at the beginning and the end of the isotopy are
normal variations of each other inside tubes of radius bounded by the injectivity
radii of the normal exponential map of the curves. Further, one can ensure that
the inverses of these injectivity radii of closed curves during the handle slide isotopies
are uniformly bounded by a reasonably growing function of n. Now it is obvious
that Lipschitz constant of diffeomorphisms corresponding to the individual steps of
the discretized isotopy are bounded by a reasonably growing function of n.
Now, it remains to check that the number of small steps in the discretization is
also bounded by a reasonably growing function of n. In other words, the isotopies
do not need to be too long. In order to see this we can just analyze the isotopies corresponding to each of the elementary operations. (Alternatively, one can
use an argument that provides an explicit upper bound for the number of
points in a minimal ε-net in the space of Lipschitz curves of
bounded length with injectivity radius of the normal exponential map bounded
below by a positive parameter.)

The desired family of 2-knots are ϕn(Sn) in the standard
round S4. One can also perform a stereographic projection from a point on S4
far from
ϕn(Sn) and obtain desired 2-knots in R4.

4. Filling functions.

Now we are going to use a concept from [Gro]: For each Riemannian manifold, or,
more generally, length space X if finite diameter we define its filling length function
filllengthX(x) as follows. For each positive x
let Cx denote the set of all contractible
closed curves on X that have length ≤x. For each γ∈Cx
let H(γ) be the set of all homotopies contracting γ to
a point. We consider elements of
H(γ) as one-parametric families of closed curves starting at γ
and ending at a point. For each h∈H(γ) let fill(h) denotes the maximal length of a closed curve in h. Then define filllengthX(γ)
as the infh∈H(γ)fill(h). Finally, we define filling
Fl(X) of X as the supremum of filllengthX(γ)/length(γ) over all closed
contractible curves γ.
In order for this supremum to be finite it is helpful if all sufficiently short closed curves can be contracted to a point
without length increase as it happens, for example, for Riemannian manifolds. Further, note that if X is, in addition, simply
connected, then Fl can be majorized in terms of the value Fl≤2diam(X)(X) of this supremum on curves of length ≤2diam(X).

The choice of 2diam(X) here is due to the well-known fact ([Gr]) that
each closed curve γ in a Riemannian manifold (possibly with boundary) X can be
homotoped with almost no increase of length to a join of closed curves of length ≤2diam(X) and contracted through closed curves of length ≤
length(γ)+ constdiam(X). Indeed, one can choose any point z∈X, a
finite set of points x1,…xN on γ such that xi and xi+1 are close
to each other, and to reduce contracting γ to contracting geodesic triangles
zxixi+1. We see that if X is simply-connected, then the length of γ will increase by at most
4diam(X)+Fl≤(2diam(X)+ε)(X)(2diam(X)+ε) for the described contracting homotopy, and then one can pass to the limit as ε⟶0.

In this paper we are going to apply this concept to spaces that are not simply connected, but have fundamental groups isomorphic to Z.
These spaces will be complements to trivial 2-knots, and it is easy to see that the generators of Z can be represented by based loops of length
that does not exceed twice the diameter of these spaces plus an arbitrarily small ε. Indeed,one can take a very small circle around
the embedded S2 and connect it with the base point by two minimizing geodesics travelled in the opposite directions. Denote
the resulting curve by τ. Now proceed as in the simply connected case with the only difference that distances between points xi and xi+1 on
γ in the metric of γ are now chosen as 12diam(X) rather than a very small ε. In this way we control
the number of the triangles zxixi+1 in terms of length(γ)diam(X). Each of these triangles has length <3diam(X) and is
homotopic to a point or τ which is possibly iterated several times and maybe also travelled in the opposite direction. Once we homotope
γ into a collection of integer iterates of τ (where the exponents must sum to zero), we will be able to cancel them and contract
the resulting curve without increasing its length. Therefore, in order to prove the finiteness of Fl it is sufficient to demonstrate
the existence of the supremum of the maximal length of loops in an “optimal” homotopy contracting a loop of length ≤3diam(X) to an integer
power of τ. (Here the supremum is taken over all loops of length ≤3diam(X); the word “optimal” means that we are taking
the infimum over all such homotopies; the power of τ is uniquely determined by the initial curve and is locally constant). The existence
of this supremum becomes evident when we combine the following two facts: First, note
that the Ascoli-Arzela theorem implies the compactness of the set of closed curves in X of bounded length parametrized proportionally to the
arclength. Second, assume the existence of δ>0 such that each closed loop of length ≤δ can be contracted to a point
without length increase. How let γ1, γ1 be two loops such that for each td(γ1(t),γ2(t))≤δ.
Then γ1, γ2 will be homotopic to the same power of τ, and given a homotopy between γ1 and this power of τ,
we can extend it to a homotopy for γ2 by merely adding a homotopy between γ2 and γ1 that does not increase length
by much. This implies the second
fact that the maximal length of loops in an “optimal” isotopy cannot significantly increase under (controllably) small perturbations of loops.

Now we plan to use these filling functions similarly to how it had been done in [N2]. The main idea is that
they behave in a similar way to their algebraic counterparts that measure how difficult it is to see that all trivial
words in a “visible” finite presentation of the fundamental group of X are, indeed, trivial (or, more concretely, the
maximum over all trivial words of a given length of the minimal area of a van Kampen diagram for the considered word).
More specifically, we would like to consider complements to Sn in M(Qn), ϕn(Sn) in the standard S4 and to establish that
1) the values of Fl for these complements are similarly growing functions of n; 2) Fl for the complements of Sn in M(Qn) is a
not reasonably growing function of n, as its growth is more or less the same as the growth of the area of van Kampen diagrams required
to demonstrate that the groups of these 2-knots are trivial (see Theorem 2.1); and 3) If knots ϕn(Sn) in the standard S4
can be untied through 2-knots of not too high complexity, then the second of these two filling functions is reasonably growing.
Taken together these three facts establish that the constructed 2-knots can be untied only through 2-knots of a very high complexity.
One technical difficulty that arises here is the following: As the considered complements to 2-sphere are not
compact, it is not clear that the values of Fl for the considered complements are finite. More specifically, one can have contractible
curves in, say, M(Qn)∖Sn that include many very short arcs that go around Sn in opposite directions.
Also, the proof of the existence of Fl given above used the existence of δ such that all closed curves of length ≤δ are contractible
(and even contractible by a length non-increasing homotopy).
Therefore, we are going to remove not only the 2-knots but also their open tubular neighborhoods with radii given as the inverse
values of a reasonably growing
function of n. More specifically, we proceed as follows.

In order to establish that ϕn(Sn) cannot be untied through 2-knots of a not too high complexity
we proceed by contradiction. We assume that ϕn(Sn) in the standard S4 can be isotoped to the standard unknot
through 2-knots of complexity that is a reasonably growing function of n. Consider the smooth case, when the complexity
is defined as √Arear, where r is the injectivity radius of the normal exponential map. (The proof in the simplicial
case is quite similar.) First, we observe that that there is a reasonably growing function f(n) that majorizes √Arear
not only for ϕn(Sn) in the standard S4 and Sn in M(Qn)∖Sn but also for all 2-knots in an isotopy
connecting ϕn(Sn) with an unknot in the round S4. (This fact follows from our assumption that ϕn(Sn) can be untied through knots of not very high complexity.) Without any loss of generality
we can normalize all metrics and assume that the areas of Sn, ϕn(Sn), and all knots in an isotopy of ϕn(Sn) to a round
2-sphere in a round 4-sphere are between one and g(n), and also
1F(n) is a lower bound for the injectivity radii of all these 2-knots,
where F and g are some reasonably growing functions of n.
For each of these 2-knots k let N(k) denote its open tubular neighborhood of radius 110F(n). We modify our idea and consider
the complements to N(Sn) in M(Qn) and N(ϕn(Sn)) in a round S4 rather than to Sn and ϕn(Sn). In this way
we obtain compact metric spaces and immediately see that their Fl are finite. Moreover, the boundaries of these
spaces are hypersurfaces (diffeomorphic to S2×S1) with principal curvatures bounded by a reasonably growing function of n.
The same will hold also for the complements of N(kt) in the round S4, where kt denote the 2-knots in the considered isotopy
between ϕn(Sn) and a round 2-sphere. Now we are going to prove the following three lemmae:

Lemma 4.1.

The functions Fl(M(Qn)∖N(Sn)) and Fl(S4∖N(ϕn(Sn))) are similarly
growing functions of n, where S4 denotes the standard round sphere.

Proof.

The assertion of the lemma immediately follows from the fact that Lipschitz constants
of ϕn and its inverse are bounded by reasonably growing functions of n. (We explained this fact
at the end of the previous section.) This easily implies that M(Qn)∖N(Sn) and S4∖N(ϕn(Sn))
are also bi-Lipschitz homeomorphic with both Lipschitz constants bounded by reasonably growing functions.
∎

Now we are going to prove that:

Lemma 4.2.

Fl(M(Qn)∖N(Sn)) is not reasonably growing.

Proof.

The idea of our proof of Lemma 4.2 is that Fl(M(Qn∖N(Sn))) behaves essentially
as the Dehn function(s) for the family of finite presentations Pn.
More precisely, assume that Fl(M(Qn)∖N(Sn)) is reasonably growing.
Then we are going to prove that there exist van Kampen diagrams for xn
in Pn with C(n) cells, where C(n) is a reasonably growing function.
This will contradict Theorem 2.1.

Let us respresent xn by a sufficientlyly short loop in M(Qn)∖N(Sn)
and contract it to a point via loops of length ≤l(n), where l
is a reasonably growing function of n. Each of these intermediate loops can be
first homotoped to a loop in the 2-skeleton of Fl(M(Qn)∖N(Sn)) that can be regarded as the
2-dimensional Dehn complex corresponding to the
finite presentation Pn, and afterwards
almost canonically represented as the product by ≤constl(n) of loops representing the generators
of Pn. “Almost canonically” means that the only ambiguities
appear as the result of
having different ways to represent the same loop γ by a small number of short words that correspond to relators of Pn.
The number of these small words is proportional to the length of γ and is, therefore, bounded by a reasonably growing function of n.
These amiguities will correspond to
the discontinuities in our presentations of loops by words, and together will provide a representation of xn as the product
of conjugates of words corresponding to these ambiguities.
As the length of words
is also bounded by a reasonably growing function of n, so will be the number of cells
in the corresponding van Kampen diagram. This completes the proof of Lemma 4.2. (Note that this argument is very similar
to an analogous argument in the proof of Proposition 2.1 in our paper [LN].)
∎

Finally, we are going to prove that:

Lemma 4.3.

If the constructed 2-knots
in S4 can be untied through 2-knots of complexity bounded by
a reasonably growing function, then
Fl(S4∖N(ϕn(Sn))) is a reasonably
growing function.

It is clear that
Lemmae 4.1, 4.2 and 4.3 together immediately yield the contradiction that implies that the constructed family of
2-knots satisfies the conditions of our main theorem.

Proof.

To prove Lemma 4.3 we assume that the constructed trivial knots can be untied through knots kt,t∈[0,1]
of complexity bounded by a reasonably growing function. As we are considering the smooth case, we can assume that
the areas of the knots during the homotopy are between 1 and g(n), where g is a reasonably growing function, and the injectivity radius
of the normal exponential map during a contracting isotopy is bounded below by
1F(n), where F is also a reasonably growing function of n.

Now our goal is to demonstrate that any closed curve γ of length
l≤1100F(n) in S4∖kt can be contracted with an increase of length bounded by a constant factor (say, 5). If γ is not
in l-neighborhood of ∂N(kt), then it is a convex metric ball of radius l2 in S4 and can be contracted
within this ball without any length increase. Otherwise, it can be homotoped along outer normals to ∂N(kt) to the outer
boundary of the l-neighborhood of ∂N(kt) (or, equivalently, (110F(n)+l)-neighborhood of kt). Note that
the length of each arc of γ under the projection increases less than by a factor of 2. It is easy to homotope γ to its
projection that we denote ~γ through curves of length less than 5l. (First, we continuously grow two copies
of the normal between a point of γ
and its projection on ~γ. Then we start moving one of the copies of the normal by moving its endpoints along γ and,
correspondingly, ~γ. The intermediate curves consist of a constantly shrinking arc of γ, an expanding
arc of ~γ and the two normals. Finally, this connecting normal returns to the original segment, and we cancel two copies of
the original segment.) Now, one can contract ~γ inside a convex metric ball of radius l in S4 without a length increase.

This argument applies to the complement of 110F(n)-neighborhood of any 2-knot k in S4 such that its area
and the injectivity radius of the normal exponential map satisfy the same bounds as the bounds for kt. Now we can adapt
the argument from [N2] to prove that if two such 2-knots k1 and k2 with the injectivity radius of the normal exponential map greater
than 1F(n) and volume between 1 and g(n) are 11000F(n)-close, then the values of Fl for the complements
of N(ki),i=1,2, differ from each other by not more than a constant factor
(say, 106). The idea that in order to contract a contractible closed curve γ1 in, say, the complement to N(k1), one can transfer the curve to (the close in Gromov-Hausdorff metric) complement of N(k2) without a significant length
increase, contract the resulting curve γ2 there , discretize the contracting homotopy, transfer it back to the complement of N(k2) and “fill”
the discretized homotopy. In order for this program to work one need to be able to contract without a significant
length increase “short” closed curves of length that does not exceed the distance between the complements to N(ki) times an appropriate
constant. Note that in order to see that γ2 is contractible in the complement of N(k2) we can construct
a contracting homotopy by similarly “transfering” a homotopy contractiung γ1 in the complement to N(k1). See [N2] for detailed
descriptions of such transfers. This argument works, for example, if the distance between ∂N(ki),i=1,2, does not exceed
110000F(n).

Now note that
the isotopy between the standard unknot and given unknot can be replaced by a
sequence of “jumps” of “length” ≤1100000F(n) where the number
of jumps is bounded by a reasonably growing function of n. Here “length” means the Hausdorff distance between the considered knots.
The number of these jumps is bounded by twice the number of pairwise
disjoint metric balls of radius
1100000F(n)-net in the considered space of hypersurfaces in S4 satisfying the same bounds for the volume
and lower bounds (110F(n)) for the injectivity radius of the normal exponential map as ∂N(kt) for 2-knots kt. Indeed,
we can replace any subsequence of “jumps” where the distance between the beginning
and the end does not exceed 1100000F(n) by just one jump.

Now note that sizes of constF(n)-nets in the space
of hypersurfaces in S4 with areas between 1 and a reasonably growing function, and the injectivity radius of the normal exponential map bounded below
by the inverse of a reasonably growing function
is also bounded by a reasonably growing function.
Such a bound will follow from any proof of the precompactness
of the corresponding space of of hypersurfaces of bounded complexity and volume (cf. [N]). One possible idea is to represent
these hypersurfaces as zero sets of appropriate C2-functions that vary in the same way along each normal segment to the hypersurface
from −1 and 1. It is easy to majorize
C2-norms of these functions in terms of the available data, and to use a standard effective proof of the Ascoli-Arzela theorem (see [N]
for the details of this argument).

Since the change of Fl of the complements to the neighborhoods of 2-knots under such jumps does not exceed an explicit constant factor (cf. [N2], [LN]),
we can start at the standard unknot , “jump” back to the given 2-knot and observe
that Fl for the complement ito its open neighbourhood of radius 110F(n)
does not exceed const#jumps, which is bounded by a reasonably growing function of n.

Note that one can do a similar argument for 2-knots in R4 instead of S4, but in order to have the desired compactness
one needs to transfer all 2-knots during isotopies to a neighborhood of the origin (by appropriate translations).
∎

Again, one can easily adopt this proof for the PL -case.

Acknowledgements. This research has been partially supported by NSERC Accelerator and Discovery Grants of A. Nabutovsky.